Universal kinetics in reaction-limited aggregation

Reaction-limited cluster aggregation is modeled with the kinetic rate (Smoluchowski) equations, using a kernel determined intrinsically by the clusters fractal geometry. The kernel scales with cluster mass as ${\mathrm{M}}_1$ ${\mathrm{M}}_2^{\lambda \mathrm{-}1}$ (${\mathrm{M}}_1$≫${\mathrm{M}}_2$), and ${\mathrm{M}}_1^{\lambda }$ (${\mathrm{M}}_1$≊${\mathrm{M}}_2$), with λ=1 in three dimensions, resulting in exponential kinetics and a cluster mass distribution ${\mathrm{C}}_{\mathrm{M}}$∼${\mathrm{M}}^{\mathrm{-}\mathrm{\tau }}$, with τ=(3/2), in excellent accord with experiments. The singular nature of this solution forces the adjustment of the cluster fractal dimension, ${\mathrm{d}}_{\mathrm{f}}$, thereby determining its value.